ACTAUNIVERSITATIS

UPSALIENSISUPPSALA

2017

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 1532

Location and Relocation of SeismicSources

KA LOK LI

ISSN 1651-6214ISBN 978-91-513-0013-9urn:nbn:se:uu:diva-327038

Dissertation presented at Uppsala University to be publicly examined in Hambergsalen,Geocentrum, Villavägen 16, Uppsala, Friday, 15 September 2017 at 10:00 for the degreeof Doctor of Philosophy. The examination will be conducted in English. Faculty examiner:Professor Lars Ottemöller (University of Bergen, Department of Earth Science).

AbstractLi, K. L. 2017. Location and Relocation of Seismic Sources. Digital ComprehensiveSummaries of Uppsala Dissertations from the Faculty of Science and Technology 1532. 73 pp.Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-513-0013-9.

This dissertation is a comprehensive summary of four papers on the development and applicationof new strategies for locating tremor and relocating events in earthquake catalogs.

In the first paper, two new strategies for relocating events in a catalog are introduced.The seismicity pattern of an earthquake catalog is often used to delineate seismically activefaults. However, the delineation is often hindered by the diffuseness of earthquake locationsin the catalog. To reduce the diffuseness and simplify the seismicity pattern, a relocation anda collapsing method are developed and applied. The relocation method uses the catalog eventdensity as an a priori constraint for relocations in a Bayesian inversion. The catalog event densityis expressed in terms of the combined probability distribution of all events in the catalog. Thecollapsing method uses the same catalog density as an attractor for focusing the seismicity inan iterative scheme. These two strategies are applied to an aftershock sequence after a pair ofearthquakes which occurred in southwest Iceland, 2008. The seismicity pattern is simplifiedby application of the methods and the faults of the mainshocks are delineated by the reworkedcatalog.

In the second paper, the spatial distribution of seismicity of the Hengill region, southwestIceland is analyzed. The relocation and collapsing methods developed in the first paper and anon-linear relocation strategy using empirical traveltime tables are used to process a catalogcollected by the Icelandic Meteorological Office. The reworked catalog reproduces details ofthe spatial distribution of seismicity that independently emerges from relative relocations of asmall subset of the catalog events. The processed catalog is then used to estimate the depth tothe brittle-ductile transition. The estimates show that in general the northern part of the area,dominated by volcanic processes, has a shallower depth than the southern part, where tectonicdeformation predominates.

In the third and the fourth papers, two back-projection methods using inter-station crosscorrelations are proposed for locating tremor sources. For the first method, double correlations,defined as the cross correlations of correlations from two station pairs sharing a commonreference station, are back projected. For the second method, the products of correlationenvelopes from a group of stations sharing a common reference station are back projected. Backprojecting these combinations of correlations, instead of single correlations, suppresses randomnoise and reduces the strong geometrical signature caused by the station configuration. Thesetwo methods are tested with volcanic tremor at Katla volcano, Iceland. The inferred sourcelocations agree with surface observations related to volcanic events which occurred during thetremor period.

Keywords: Earthquake location, Earthquake relocation, Tremor location, Bayesian inversion,Correlation, Earthquake catalog, Iceland, Seismicity, Interferometry, Inverse theory, Volcanictremor

Ka Lok Li, Department of Earth Sciences, Geophysics, Villav. 16, Uppsala University,SE-75236 Uppsala, Sweden.

© Ka Lok Li 2017

ISSN 1651-6214ISBN 978-91-513-0013-9urn:nbn:se:uu:diva-327038 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-327038)

我曾專心用智慧考察研究過天下所發生的一切；這實在是天主賜與人類的一項艱辛的工作。

訓道篇 1:13

Wisely I have applied myself to investigation and exploration of everythingthat happens under heaven. What a wearisome task God has given humanity

to keep us busy!

Ecclesiastes 1:13

List of papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I Li, K. L., Ó. Gudmundsson, A. Tryggvason, R. Bödvarsson, and B.Brandsdóttir (2016), Focusing patterns of seismicity with relocationand collapsing, Journal of Seismology, 20(3), 771-786,doi:10.1007/s10950-016-9556-x.

II Li, K. L., C. Abril, Ó. Gudmundsson et al. (2017), Seismicity of theHengill area, SW Iceland: Details revealed by catalog relocation andcollapsing, Manuscript to be submitted.

III Li, K. L., G. Sgattoni, H. Sadeghisorkhani, R. Roberts, and O.Gudmundsson (2017), A double-correlation tremor-location method,Geophysical Journal International, 208(2), 1231-1236,doi:10.1093/gji/ggw453.

IV Li, K. L., H. Sadeghisorkhani, G. Sgattoni, O. Gudmundsson, and R.Roberts (2017), Locating tremor using stacked products ofcorrelations, Geophysical Research Letters, 44(7), 3156-3164,doi:10.1002/2016GL072272.

Reprints were made with permission from the publishers. In addition, belowis a list of other publications during my PhD studies which are not included inthis thesis.

• Sgattoni, G., Ó. Gudmundsson, P. Einarsson, F. Lucchi, K. L. Li, H.Sadeghisorkhani, R. Roberts, and A. Tryggvason (2017), The 2011 un-rest at Katla volcano: Characterization and interpretation of the tremorsources, Journal of Volcanology and Geothermal Research, 338, 63-78,doi:10.1016/j.jvolgeores.2017.03.028.

• Sadeghisorkhani, H., Ó. Gudmundsson, K. L. Li, A. Tryggvason, R.Roberts, K. Högdahl, and B. Lund (2017), Surface wave tomography ofsouthern Sweden from ambient seismic noise, Manuscript to be submit-ted.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Part I: Earthquake location and relocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 The basics of earthquake location and relocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1 The earthquake location problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Absolute location methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1 Non-linear methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.2 Iterative linearized inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.3 Bayesian inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Relative location methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4 Uncertainty estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5 Earthquake relocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5.1 Relocation using empirical traveltime tables . . . . . . . . . . . . . . . . . 242.6 The scatter of earthquake locations in a catalog . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6.1 The collapsing method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.6.2 The condensation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Summary of Paper I and II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.1 Summary of Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1.1 The relocation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.1.2 The collapsing method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1.3 The May 2008 aftershock sequence in Iceland . . . . . . . . . . . . . 30

3.2 Summary of Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.1 Tectonic setting and seismicity of Hengill . . . . . . . . . . . . . . . . . . . . . 343.2.2 Results of relocations and collapsing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2.3 Estimating the depth to the brittle-ductile transition . . . . . 36

Part II: Tremor location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 Tremor source location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.1 Tremor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1.1 Volcanic tremor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.1.2 Nonvolcanic tremor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2 Tremor location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2.1 Amplitude decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2.2 Semblance method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2.3 Method of correlation coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3 Back projection of cross correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5 Summary of Paper III and IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.1 Summary of Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1.1 The double correlation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.1.2 Volcanic tremor at Katla volcano, Iceland . . . . . . . . . . . . . . . . . . . . . 535.1.3 Synthetic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.2 Higher-order cross correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.3 Summary of Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.3.1 Tremor location using products of correlationenvelopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.3.2 Synthetic and real-data examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.4 Tremor location using a probabilistic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6 Sammanfattning på svenska (Summary in Swedish) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

7 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

1. Introduction

A seismic source is any device or process that causes mechanical disturbanceto the ground and therefore generates seismic waves that propagate throughthe Earth to the recording seismographs. There are different types of sourcesthat produce seismic energy, e.g., earthquakes, explosions, human activities,ocean waves, volcanic processes etc.

Identifying the locations of seismic sources is usually the first task when an-alyzing the seismic waves from uncontrolled sources. It is a critical step inthe workflow of the analysis since a wrong estimate of the source locationwill result in wrong interpretations in the subsequent analysis. For example,in a volcanic area, a tremor source that is being located near the surface mayrequire a very different model to describe the source generation mechanismcompared to a source being located at depth. Besides, the locations of seismicsources may reveal information about the geology near their hypocenters. Forexample, it is common in seismology to use earthquake seismicity to delineatefaults in an area. Having accurate locations will therefore allow us to betterunderstand the structures of the fault systems.

For the purpose of source location, it is useful to classify seismic sources intotwo main groups based on their recorded waveforms in the seismograms, 1)sources with clear onset times and 2) sources without a clear onset time. Forthe first type of sources, one can clearly identify the impulsive seismic phasesin the waveforms and therefore pick the onset times of those phases. Sourceslike earthquakes and explosions belong to this type. For the second type ofsources, there is usually no identifiable phase expect for a continuous recordof vibrations in the seismogram. Therefore, it is impossible to pick an onsettime for these sources. Examples for this type of sources include tremor andemergent events.

In the location problem, whether one can pick the onset time of a seismicphase is an important factor when deciding what method to be used for lo-cating the source. If the onset times can be picked, one can use the so-calledpicked arrival-time methods. In this case, the location problem can be posedas an inverse problem which seeks a point in space and time such that the nu-merically predicted onset times fit the observed ones. However, if the onsettimes are not available, the above methods fail and one has to seek a methodthat does not rely on arrival picks. Extracting differential traveltime measure-ments based on waveform similarity from different seismic stations is one of

9

the possible and popular solutions. Strategies like the semblance method andinter-station cross correlations belong to this class. One can also make useof the fact that the amplitude of a seismic wave decays with distance in orderto deduce the distance between the source and a seismic station. Combiningseveral observations will allow us to pinpoint a location using triangulation.

In this thesis, I will discuss the location problem related to both types ofsources. It is, therefore, divided into two parts. In part I, I will discuss thelocation problem related to sources with clear onsets, that is the earthquakelocation problem. In part II, I will move on to the topic of locating sourceswith unclear onsets, i.e. tremor source location. Each part of the thesis be-gins with a chapter which introduces the nature of the problems and reviewssome of the existing methods for solving the problems. It is then followedby a chapter which summarizes papers related to the corresponding problems.This comprehensive summary includes four papers. Paper I and II are relatedto earthquake locations and relocations while Paper III and IV are associatedwith tremor locations.

In Paper I, we introduce two new strategies for earthquake relocations andreprocessing. The first strategy uses the catalog event density as an a prioriconstraint for relocations using a Bayesian inversion. The method first definesa probability distribution of an event in an earthquake catalog using its loca-tion and uncertainty estimates. The catalog density is defined as the sum of theprobability distributions of all events in the catalog except the one that is beingrelocated. This catalog density is used as an a priori constraint in a Bayesianinversion scheme and the posterior distribution is the product of the catalogdensity and the probability distribution of the current event. The event is thenmoved to the maximum of the posterior distribution. In the second method,the same catalog density is used as an attractor for focusing the seismicity inan iterative scheme. These two methods focus and simplify the seismicity pat-tern in a catalog and therefore facilitate the delineation of faults and structuresin the study area. Synthetic examples and a real-data example taken from anaftershock sequence after an earthquake doublet which occurred in Iceland areused to demonstrate the methods.

In Paper II, the spatial distribution of seismicity in the Hengill region, south-west Iceland is analyzed using the relocation and collapsing methods devel-oped in Paper I and a non-linear relocation strategy using empirical travel-time tables (Abril and Gudmundsson, 2017). The Hengill area is a triple junc-tion with volcano-tectonic activity associated with rifting, tectonic activity ona transecting transform. Significant induced seismicity has occurred due todrilling and injection of fluid into geothermal fields. Since the installation ofthe South Iceland Lowland (SIL) network in 1990 (Stefánsson et al., 1993),the Icelandic Meteorological Office has complied over 130,000 events within

10

the Hengill area over the past 20 years. In Paper II, this catalog is used toanalyze the seismicity in Hengill. Three different relocation and collapsingmethods are applied sequentially to the catalog. First, the SIL events are relo-cated using empirical traveltime tables and a non-linear, grid-search locationstrategy. Then, the events are relocated using a Bayesian inversion, where thecatalog density is used as an a priori constraint. Finally, the same catalog den-sity is used in an iterative collapsing scheme to find an alternative distributionof the seismicity. Paper II shows that the processed catalog reproduces detailsof the spatial pattern of seismicity that independently emerges from relativerelocations of a small subset of the catalog events (Bessason et al., 2012).With this support, the processed catalog is used to estimate the depth extentof the seismogenic zone, associating that with the depth to the brittle-ductiletransition. The estimations show that in general the depth to the brittle-ductiletransition is shallower in the northern part than the southern part of the Hengillarea. This can be explained by the fact that the northern part is dominated byvolcanic processes while tectonic deformation predominates in the southernpart.

Paper III proposes a new method to locate tremor using inter-station cross cor-relations. This method is a back-projection method which migrates the cross-correlation functions between station pairs from the time domain to the spatialdomain. However, instead of back projecting the correlations, we back projectthe double correlation, which is defined as the cross correlation of correla-tions from two station pairs sharing a common reference station. This doublecorrelation suppresses random noise beyond the single correlation due to theincrease of the level of redundancy in the data. It also reduces the strong sig-natures of the station geometry in the back-projected correlations. Therefore,correlated noise is suppressed with the double correlations. These advantagesallow us to obtain a better estimate of the tremor location. Synthetic testswith simulations of noise processes and realistic wave propagation effects arecarried out and the double-correlation method is applied to volcanic tremor atKatla volcano, Iceland. The inferred location agrees with surface observationsrelated to volcanic events which occurred during the tremor period.

Because of the success in suppressing noise using double correlations, it isreasonable to consider locating tremor using higher-order correlations in orderto further suppress noise. Paper IV suggests another strategy which resemblesthe use of higher-order correlations, but it is much more computationally ef-ficient than calculating the full high-order correlations. In this method, wepropose to multiply together the back-projected correlation envelopes froma group of n stations sharing a common reference station. The result willbe a (n− 1)th order product of correlation envelopes. Given a network ofN stations, one can, therefore, construct a series of products with increasingorder up to the maximum (N−1)th order. With synthetic and real-data exam-

11

ples, we show that this way of combining information provides much bettersuppression of correlated noise than stacking back-projected correlations andtherefore significantly improves the location estimate.

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Part I:Earthquake location and relocationThe first part of the thesis covers the topic of earthquake location and reloca-tion, which is the theme of Paper I and II. It is divided into two chapters. Inchapter 2, I introduce the basics of the earthquake location problem and therelocation of events in earthquake catalogs. In chapter 3, I provide a summaryof Paper I and II.

2. The basics of earthquake location andrelocation

This chapter describes the basics of the earthquake location problem. It startswith a description of the problem, followed by an introduction to commonmethods that are used to solve the problem. The estimation of location un-certainty is then discussed. Finally, the chapter ends with a discussion aboutthe relocation of events in an earthquake catalog. In particular, I bring up aproblem about the scatter of event locations in an earthquake catalog. Thisproblem motivates the development of new methods described in Paper I.

2.1 The earthquake location problemEarthquake location is to determine the space and time of occurrence of theenergy released from a seismic event. A location is called “absolute location”if the location of an event is determined with respect to a fixed geographiccoordinate system and a fixed time standard. A location is called “relative lo-cation” if an event is located with respect to another event which may have anuncertain absolute location.

In a homogeneous medium with constant velocity v for a given wave type(P-wave or S-wave), the arrival time at a location (x, y, z) for a source locatedat (x0, y0, z0) and time t0 is given by

tarr = t0 +1v

[(x− x0)

2 +(y− y0)2 +(z− z0)

2]1/2

. (2.1)

Eq. (2.1) shows that a change in the spatial coordinates of the source results ina non-linear change in the arrival time. For a medium with a general velocityfield v(x), the arrival time can be expressed as an integration along the ray pathS, i.e.

tarr = t0 +∫S

u(x)dx, (2.2)

where u(x) = 1/v(x) is the slowness field of the medium and x is the spa-tial position. Eq. (2.2) is non-linear since a change of the source locationwill result in change of the ray path along which the integration is evaluated.

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Therefore, the location problem is inherently non-linear.

In addition, in an inhomogeneous medium, there can be energy exchange be-tween wave types (e.g. P to S conversions), but eq. (2.2) is only valid for asignal traveling entirely as one wave type.

The earthquake location can be formulated as an inverse problem. Assum-ing we have N observations of arrival times for some seismic phases (P or S),we can construct a data vector dobs such that

dobs =[t1arr, t2

arr, ... , tNarr

]T, (2.3)

where t iarr is the observed arrival time for the ith phase. Similarly, we can

represent our unknowns as a model vector m, i.e.

m =[x, y, z, t

]T, (2.4)

where x, y, z and t are the spatial position and the origin time of the earthquake,respectively. Then, we can write eq. (2.2) in form of

dobs = g(m), (2.5)

where g(m) is a non-linear function which predicts the arrival times for a givenm.

2.2 Absolute location methodsThe most common way to locate a seismic event is to determine the misfitbetween observed and theoretical arrival times. The observed arrival times areobtained either by manual or automated picking of phases in the seismograms.The theoretical arrival times of the corresponding phases are calculated basedon a velocity model suitable for the scale of the problem. Some norm is thenused to quantify the misfit between the observed and the theoretical arrivals.For example, if the L2 norm is chosen, the misfit E is then defined as thesquare of the L2 norm, i.e.

E =N

∑i=1

(t iobs− t i

pre)2, (2.6)

where t iobs and t i

pre are the observed and predicted arrival times for the ithphase, respectively and N is the total number of phases. The location (and theorigin time) which minimizes the misfit is then chosen to be the location ofthe event. This is the so-called least squares solution to the location problem.

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Since the location problem has only four unknowns, three for space and onefor time, and there are usually far more independent observations than theunknowns, the problem is often over determined. For this reason, it is notcommon to apply regularization to the problem.

2.2.1 Non-linear methodsAs shown in eq. (2.1), the location problem is non-linear even for the sim-plest case of a uniform velocity model, but since there are only a few un-knowns, non-linear methods (methods that do not linearize the problem) areoften quite tractable. Non-linear methods offer some important advantagesover linearized inversions. For example, non-linear methods like Monte Carlomethods work by directly sampling the model space. They do not depend on agradient approximation of the misfit function and therefore they do not rely onthe misfit function being smooth. Non-linear methods usually do not involveany potentially unstable calculation such as matrix inversion, and in this sensethey are computationally stable. With linearized inversion, it is possible thata solution is not found due to complete failure of the iterative process. How-ever, for strategies like Monte Carlo methods, even though finding an optimalsolution may be slow, a solution will be found given sufficient sampling of themodel space.

To locate an earthquake in a non-linear sense, the simplest way is to applya regular and deterministic search of the four dimensional model space andchoose the point in the model space that minimizes the misfit between the ob-served and the predicted data. Methods that adopt this concept include gridsearch and nested grid search approaches (e.g. Sambridge and Kennett, 1986;Shearer, 1997; Dreger et al., 1998). Another approach is to randomly sam-ple the model space, e.g. Monte Carlo methods (Sambridge and Mosegaard,2002). The simplest form of a Monte Carlo method is to apply uniform sam-pling to each dimension in the model space. Since uniform sampling, by defi-nition, is not biased towards any particular region of model space, it allows usto estimate the complete misfit function without any risk of being entrappedin local minima of the misfit function. However, uniform sampling is com-putationally demanding, especially for problems with large model spaces ortime-consuming forward calculations. To increase the efficiency of a MonteCarlo method to estimate a misfit function, one can choose a sampling densitywhich follows closely to the misfit function. Methods that follow this rule arereferred to as importance sampling methods (Peter Lepage, 1978; Mosegaardand Tarantola, 1995), e.g. Metropolis-Hastings algorithm (Hastings, 1970).

Directed, stochastic search method is another class of methods which is com-monly used in earthquake locations (e.g. Kennett and Sambridge, 1992; Billings,

17

1994). Unlike methods with uniform sampling or importance sampling, di-rected search methods in general do not explore the model space in a mannerthat can produce a complete misfit function. Instead, they are designed tosearch for the global minimum of the misfit function possessing a large num-ber of secondary minima. Examples for this class of methods include simu-lated annealing (Kirkpatrick et al., 1983), genetic algorithm (Holland, 1975)and neighborhood algorithm (Sambridge, 1999).

2.2.2 Iterative linearized inversionAlthough non-linear methods offer many advantages over linearized inver-sions, surprisingly many earthquake-monitoring networks still employ a lin-earized inversion scheme (SNSN, 1904; Stefánsson et al., 1993). As discussedin section 2.1, for an earthquake location, one can relate the observations ofarrival times to the hypocenter (and the origin time) of an event via

dobs = g(m), (2.5)

where the notation follows that in section 2.1. In the linearized inversion, astarting model, which is usually a “best guess” model, is first chosen. Thestarting model m0 is expressed as a vector

m0 = (x0, y0, z0, t0)T , (2.7)

where x0, y0, z0 and t0 are the spatial position and the origin time of the initialguess, respectively. We then seek an update to the model, which, to the firstorder, can be expressed as

m1 = m0 +δm0, (2.8)

where m1 is the updated model and δm0 is the change that moves the modeltowards a better fit to the data. Combining eq. (2.5) and (2.8), we have

dobs = g(m1) = g(m0 +δm0)

≈ g(m0)+∂g∂m

∣∣∣∣m=m0

δm0

= dpre +∂g∂m

∣∣∣∣m=m0

δm0, (2.9)

where dpre is the predicted arrival times for the model m0. Rearranging eq.(2.9) gives

δd =∂g∂m

∣∣∣∣m=m0

δm0, (2.10)

18

where δd = dobs−dpre. I have now linearized the problem since the differ-ence between the observed and the predicted arrival times is linearly relatedto the change of the model. Eq. (2.10) can be written as a system of linearequations which can be solved iteratively for an initial guess of the model m0.The idea is that the model iteratively moves towards a minimum guided by thelocal gradient of the misfit function at the current position of the model. Thislinearized inversion is sometimes called the Geiger’s Method (Geiger, 1910).

This method is easy to implement and is computational efficient. However,there may be a risk that the final solution is only a local minimum, not a globalminimum of the misfit function.

2.2.3 Bayesian inversionBayesian inversion is another major class of methods that is often used inearthquake locations (e.g. Hirata and Matsu’ura, 1987). In this formulationthe solution to the location problem is given by the posterior probability dis-tribution Pf (m) as a function of the model m over the model space. Thisposterior distribution contains all the information available on the models,which originates from the data and the data-independent prior information.The information from the data is given by a likelihood function L(m), whichmeasures the likelihood of a model m through the misfit function E(m). Theprior information, expressed as an a priori probability density Pa(m), pro-vides knowledge of the model from, e.g., earlier observations or measure-ments (Martinsson, 2013; Li et al., 2016). The Bayesian inversion defines theposterior distribution as

Pf (m) = c1 L(m) Pa(m), (2.11)

where c1 is a constant. The “best” location is then inferred from the maximumof the posterior distribution. If there is no explicit mathematical expression forL(m) and/or Pa(m), Monte Carlo sampling can be used to explore Pf (m).

For an earthquake location problem, if we assume a Gaussian distribution forthe likelihood function, i.e.

L(m) = c2 exp{− 1

2

[d−g(m)

]TC−1

d

[d−g(m)

]}, (2.12)

where c2 is a constant, d is the data vector containing all the observations ofarrival times, Cd is the data covariance matrix holding information about thedata uncertainties, and g(m) is a function of the model m which predicts thearrival times, and a uniform probability distribution for the a priori distribu-tion, i.e.

Pa(m) = K, (2.13)

19

where K is a constant, then the Bayesian inversion will be equivalent to theleast squares method weighted by the data covariance matrix Cd.

2.3 Relative location methodsIn contrast to absolute location methods, the relative earthquake location meth-ods determine the spatial position and the origin time of an event with respectto other event(s). The main reason for applying a relative location approach isto improve the precision of the event locations. The accuracy of an absolutehypocenter location is controlled by a number of factors, e.g., the station ge-ometry, the knowledge of the velocity structure and the accuracy of the arrivaltime measurements. The effects due to an imperfect velocity model can be re-duced by using relative location methods (e.g. Poupinet et al., 1984; Frémontand Malone, 1987; Got et al., 1994). Consider two earthquakes for whichtheir hypocentral separation is small compared to the source-receiver distanceand the characteristic length of velocity heterogeneity, the ray paths from thehypocenters to a common station will be very similar for both events. It is thenpossible to use the difference in traveltimes for two events at a common stationto determine the spatial offset between the events to a high precision. This isbecause the systematic errors due to an imperfect velocity model will be can-celed out except for the part where the ray paths are not common. To furtherimprove the location precision, one can minimize the errors due to arrival timemeasurements. This can be achieved by measuring the differential times, in-stead of absolute times, with waveform correlation techniques. Assume thatthere are two closely located earthquakes with nearly identical source mech-anisms, their waveforms recorded at a common station will be very similar.Correlating the two waveforms will, therefore, extract their differential arrivaltime to a high precision. Studies have shown that it is possible to achieve a pre-cision of 1 ms for differential time measurements using waveform correlationmethods (e.g. Poupinet et al., 1984; Frémont and Malone, 1987), where theprecision for routinely picked phase onset is about 0.1 to 0.3 s. This impliesthat by using relative location methods with differential time measurements,one can calculate the relative location between earthquakes with errors of onlya few meters to a few tens of meters.

The master event approach is a commonly used method to obtain the rela-tive locations of events from differential traveltime measurements (e.g. Fré-mont and Malone, 1987; Stoddard and Woods, 1990; Deichmann and Garcia-Fernandez, 1992). In this approach, each event in a cluster is located relativeto only one event, the master event. The master event is often chosen to be anaccurately located event since an error in the location of the master event willaffect the absolute location of the whole cluster. In order for the ray paths tobe sufficiently similar so that the errors due to the velocity heterogeneity can

20

be considered to be canceled out, the events have to be sufficiently close to themaster event. This limits the spatial extension of the event cluster that can belocated.

The limitation of the master event approach can be overcome by relativelylocating events with respect to each other, instead of a single master event(Got et al., 1994). Waldhauser and Ellsworth (2000) extended this idea anddeveloped a “double difference algorithm”. Recalling eq. (2.10),

δd =∂g∂m

∣∣∣∣m=m0

δm0, (2.10)

the data residual δd is linearly related to the model perturbation δm0. Thedouble difference algorithm starts with the definition of the “double differ-ence”, which is the differential data residual between two events, i.e.

δ rki j = (t ik

obs− t jkobs)− (t ik

pre− t jkpre), (2.14)

where t ikobs and t ik

pre are the observed and predicted arrival times for the ithevent and the kth phase, respectively. Note that in eq. (2.14), I drop the vectornotation for the data vector d and write out a particular element of the vector inorder to be consistent with the notation in Waldhauser and Ellsworth (2000).Next, Waldhauser and Ellsworth (2000) generalized eq. (2.10) to describe thechange in hypocentral distance between two events i and j in relation to thedouble difference,

∂gki

∂mδmi−

∂gkj

∂mδm j = δ rk

i j, (2.15)

where gki is the function which predicts the arrival time for the ith event and

the kth phase, δmi is the model perturbation for the ith event. Eq. (2.15) isthen applied to all event pairs for a station and for all stations to form a systemof linear equations, which can be solved with some linear inversion scheme.

Slunga et al. (1995) proposed a location scheme which combines techniquesof absolute and relative locations in order to achieve accurate relative loca-tions of clusters of similar earthquakes (earthquakes with similar waveforms).Similar to Waldhauser and Ellsworth (2000), Slunga et al. (1995) defined thearrival time residual,

ea(i, j,k) = tobs(i, j,k)− tpre(i, j,k), (2.16)

and the residual of arrival time differences,

ed(i, j,k1,k2) = tdobs(i, j,k1,k2)− tpre(i, j,k2)+ tpre(i, j,k1), (2.17)

21

where tobs(i, j,k) and tpre(i, j,k) denote the observed and the predicted arrivaltimes at station i, phase j and event k, respectively. td

obs(i, j,k1,k2) is the ob-served time difference which is estimated directly from the correlation of sim-ilar waveforms. Note the slight difference in the notation compared to Wald-hauser and Ellsworth (2000). It is clear that the definition of the residual ofarrival time differences in Slunga et al. (1995) is exactly the same as the def-inition of the double difference in Waldhauser and Ellsworth (2000). Insteadof solving the linearized problem (section 2.2.2) using a least squares crite-rion [eq. (2.6)], Slunga et al. (1995) included three extra terms in the misfitfunction E, which is defined as

E =m

∑i=1

2

∑j=1

n

∑k=1

wa(i, j,k)e2a(i, j,k)

+m

∑i=1

2

∑j=1

n−1

∑k1=1

n

∑k2=k1+1

wc(i, j)[ea(i, j,k1)− ea(i, j,k2)

]2

+m

∑i=1

2

∑j=1

n−1

∑k1=1

n

∑k2=k1+1

wd(i, j,k1,k2)e2d(i, j,k1,k2)

+m

∑i=1

n−1

∑k1=1

n

∑k2=k1+1

wps(i,k1,k2)[ed(i,P,k1,k2)− ed(i,S,k1,k2)

]2, (2.18)

where wa, wc, wd and wps are the weights of each term, and m and n are thenumber of stations and events, respectively. The first term of the right handside of eq. (2.18) is the common least squares criterion for absolute locations.The second and the third terms refer to the misfit due to the difference inabsolute arrival time measurements and the relative differential arrival timemeasurements, respectively. The fourth term corresponds to misfit from thedifference in P and S arrivals. Slunga et al. (1995) argued that for cases withgood station coverage and some close stations near the events, the methodimproves not only the relative locations of similar events, but also the absolutelocation of the cluster. In Paper II, there is a comparison of results using thismethod with those using my relocation method proposed in Paper I.

2.4 Uncertainty estimationEarthquake location does not only involve the determination of the space andtime of the occurrence of an earthquake, but also their uncertainties. As dis-cussed in section 2.2.3, Monte Carlo methods with importance sampling allowus to sample the posterior distribution of a location problem. This posterior

22

distribution contains all the information available on the models, including in-formation on the uncertainty. However, to obtain an estimate of the locationuncertainty, realistic estimates of uncertainties in the observed arrival times,assuming Gaussian statistics, must be available and specified through the datacovariance matrix, Cd, in eq. (2.12). Estimating these arrival time errors is dif-ficult and often not attempted. Instead, the location uncertainties are usuallyestimated by approximating the posterior distribution with a Gaussian distri-bution in the vicinity of the final model. In other words, to fit a Gaussiandistribution to the posterior distribution around the final model. This Gaussiandistribution can then be used to estimate the location uncertainty of the finalmodel through its covariance matrix.

In cases where an iterative linearized inversion is used, the most common wayto estimate the uncertainty of the location is to first construct a data covari-ance matrix Cd using the data residuals in the last iteration of the inversion.A model covariance matrix Cm is then computed by propagating the errorscontained in the data covariance matrix through the inversion process, i.e.

Cm = ACdAT , (2.19)

where A is the generalized inverse for a particular inversion scheme. Theresulting model covariance matrix is the estimate of the covariance betweendifferent model parameters. In other words, the model covariances are esti-mated by linearizing the forward problem around the optimum location found.

There are a number of reasons for using the data residuals, instead of the mea-surement errors, to estimate uncertainties. First, it is not always possible toassign a measurement error to every arrival pick since not all the phases arepicked manually. Many of automated picking software do not have an optionto define the quality of a pick (e.g. Bodvarsson et al., 1996). Second, the mea-surement error only accounts for a part of the total error in the location prob-lem. Apart from the measurement errors, there are also computational errorsdue to, e.g., an inaccurate velocity model, simplified assumptions about wavepropagation etc. Therefore, it is more sensible to use data residuals, whichaccount for both measurement and computational errors, for uncertainty esti-mation.

Although arrival times are the data for the location problem, they are oftennot associated with an estimate of uncertainties. Instead, a weighting scheme,with different weights applied to the data residuals, is introduced in the uncer-tainty estimation. This scheme simulates some expectation of the error distri-bution, e.g., with distance or for P-wave picks versus S-wave picks. However,the weights are often guessed empirically without rigorous support.

In general, the errors in the location problem can be divided into two parts:

23

the random error and the systematic bias. Random errors can be due to e.g.measurement errors of the picking while systematic bias may arise from im-perfect knowledge about velocity structure. In the uncertainty estimation, thesystematic bias is not accounted for. Therefore, the location uncertainties arelikely to be under-estimated.

2.5 Earthquake relocationEarthquake relocation refers to a reworking of the earthquake locations in acatalog in the hope of improving the existing locations of the events. This in-cludes 1) adding new a priori information to the problem, e.g., using a morerealistic velocity model for the locations, 2) using a more sophisticated lo-cation method to relocate the events, e.g., using relative location methods,instead of absolute locations, to bring out details of the seismicity of an eventcluster, or 3) using a better phase detection algorithm in order to obtain moreaccurate measurements. Since earthquake relocation is only a revisit of thelocation problem, there is no fundamental difference between the earthquakelocation problem and the relocation problem. Every location method that Ihave described in the earlier sections is also applicable to the relocation prob-lem.

2.5.1 Relocation using empirical traveltime tablesAbril and Gudmundsson (2017) introduced a relocation strategy using em-pirical traveltime tables. For each station in a seismic network, an empiricaltraveltime table is first constructed for both P- and S-waves. The empiricaltraveltime tables are then used as an input to a nested grid-search relocationalgorithm. These empirical traveltime tables are similar to event based stationcorrections and they take into account the three-dimensional velocity structureof the area.

Abril and Gudmundsson (2017) assume that traveltimes in an earthquake cat-alog can be described as the sum of a predicted traveltime by a 1D velocitymodel, T0(x), and a residual, ∆T (x), i.e.

T (x) = T0(x)+∆T (x)= T0(x)+dTd(x)+dTr(x)+δT (x), (2.20)

where dTd(x) describes average deviations of the observed residuals from thereference velocity model (non-stationary or deterministic 3D station correc-tions), dTr(x) is a contribution of small-scale random variations of the struc-ture with respect to the velocity model, and δT (x) corresponds to the error inestimation of traveltime. The deterministic dTd term is estimated by simple

24

spatial averaging. Variograms of the remaining residuals (dTr + δT ) are thenused to estimate the variance of spatially incoherent errors (δT ) and dTr es-timated by a local fit to the remaining residuals to within the estimated errorvariance. Thus, a three-dimensional time field, Temp(x), is estimated for eachstation in the network composed of

Temp(x) = T0(x)+dTd(x)+dTr(x), (2.21)

representing traveltime between that station and any arbitrary point in spacewhich is then used as a description of the forward problem for earthquakelocation (by interpolation). The estimation of the empirical traveltime, as de-scribed above, involves estimating error variance which is used to calibratemisfit (as weight) in the earthquake location scheme. The earthquakes are re-located by a non-linear nested grid-search strategy.

This strategy is used in the relocation of the South Iceland Lowland (SIL)catalog presented in Paper II.

2.6 The scatter of earthquake locations in a catalogThe earthquake locations in a catalog are generally diffuse, i.e. they are scat-tered in space. The diffuseness may be due to errors in the location process,e.g., due to imprecise measurements of the arrival times, the use of an im-perfect velocity model or simplified theory (e.g. ray theory) to calculate thepredicted arrival times. However, the diffuseness may also be a natural prop-erty of the earthquake process. A part of the diffuseness may be random. Apart may manifest itself as a systematic bias.

Using the seismicity pattern to delineate seismically active tectonic fault zonesis a common task in seismology. However, the delineation is often hinderedby the diffuseness of the earthquake locations. Therefore, one may ask if it ispossible to reduce the scatter of the locations in order to bring out details of theseismicity pattern in a catalog. Using the relative location methods describedin section 2.3 is one of the possible solutions. By using relative location meth-ods, one can constrain earthquake locations to a high precision because dif-ferential time measurements are accurate and relative timing measurementsreduce the effects of velocity heterogeneity on the locations. However, thesemethods require precise correlation measurements of the double difference ofarrival-time residuals between two nearby events of very similar waveform(e.g. Spence, 1980; Ito, 1985; Slunga et al., 1995). This is not always possi-ble. It can only be achieved if such dense clusters exist and precise relativetiming measurements are done. In the following discussion, I concentrate onearthquake catalogs which contain absolute location estimates based on arrival

25

times alone and ask if this information can be better utilized in order to reducethe scatter of the locations and simplify the seismicity pattern.

2.6.1 The collapsing methodTo answer the above question, Jones and Stewart (1997) proposed a strat-egy called the collapsing method to reduce the scatter of earthquake locations.They let the events be attracted by their neighbors and thus obtain an alter-native distribution of seismicity with more focused locations. Each event isattracted towards the center of mass of all events within a specified confidenceinterval distance from it.

Two parameters are used in the collapsing method. First, a multiplication fac-tor for the confidence interval, which defines a three-dimensional uncertaintyellipsoid used for computing the center of mass of the neighboring events.Second, the fraction of distance between the event locations and the centroidtowards which each event moves in each iteration. In their implementation,Jones and Stewart (1997) chose to define as neighbors all events within fourstandard deviations from the current event. They chose the number 0.61803for the latter parameter, mimicking the golden section. These two parametersare artificial and chosen without any rigorous argument.

The collapsing method suffers from an artifact which can be described as clus-tering on a scale comparable to the characteristic location uncertainties of thecatalog. Jones and Stewart (1997) and Nicholson et al. (2000) demonstratedthis artifact using synthetic catalogs with a uniform distribution of events.Even if the initial distribution is uniform, the finite size of the catalog leadsto voids in the distribution on the scale of errors.

In the collapsing method, only a small portion of the catalog is used to judgehow an event is moved. In other words, the method only uses the informa-tion from a local region around the current event to decide the movement ofthat event. In paper I, we propose a modified collapsing method, which takesinto account the seismicity of the whole catalog when judging how an event ismoved. This modified collapsing method also reduces some of the artifacts inthe original collapsing method.

2.6.2 The condensation methodKamer et al. (2015) proposed a strategy, which they called the condensationmethod, to process an earthquake catalog. The motivation for this methodis slightly different from the collapsing method of Jones and Stewart (1997).Instead of simplifying the seismicity pattern by deliberately attracting events

26

towards an artificial attractor, the condensation method exploits the redun-dancy in the locations of a catalog without affecting the geometrical densityinformation they contain. It reduces the size of earthquake catalogs but avoidschanging the geometrical information in the catalogs. This is achieved bycondensing warm and vaporized events (events with large uncertainties) onto cooler, condensed events (event with small uncertainties). At the start ofthe process, the events are sorted according to their uncertainties and given aweight of unity. The weight of poorly located events is then successively trans-ferred to those neighboring events that are better located in proportion to thatpart of the target events domain where their probability density is maximalamong all overlapping events. The process continues until all events exceptthose with the lowest variance have been processed.

The condensation method has only a slight effect on the geometrical distri-bution of a catalog since it is designed to avoid that. It, therefore, has littleeffect in reducing the diffuseness of a catalog.

27

3. Summary of Paper I and II

3.1 Summary of Paper IThe study in Paper I was motivated by the problem described in section 2.6,i.e. how can one utilize the information about absolute location estimates inan earthquake catalog in order to reduce the scatter of the locations, simplifythe seismicity pattern, and therefore delineate seismically active fault zones?

Paper I introduces two new strategies to tackle this problem. The first strategyis a relocation strategy using Bayesian inversion, where the catalog density isused as an a priori constraint. I will refer to this strategy as “the relocationmethod”. The second strategy is not a relocation strategy in a strict sense, i.e.it does not minimize the misfits between observed and predicted data. It israther like an image processing method which simplifies the seismicity pat-tern. This strategy uses the same catalog density defined for the relocationmethod as an attractor for collapsing events towards a more focused distribu-tion in an iterative way. I will refer to this strategy as “the collapsing method”since it shares the philosophy of the collapsing method by Jones and Stewart(1997). In later discussion, to avoid confusion, when I refer to the collapsingmethod of Jones and Stewart (1997), I will mention it explicitly.

These two methods are applied to an aftershock sequence following the 29May 2008 earthquake doublet in southwest Iceland. The collapsing methodof Jones and Stewart (1997) and the condensation method of Kamer et al.(2015) are also applied to the aftershock sequence for comparison. In PaperI, we apply our collapsing method to synthetic data. However, the synthetictests are not discussed in this summary. Interested readers are referred to thecorresponding section in Paper I.

3.1.1 The relocation methodSince an earthquake is located as a set of spatial coordinates with some un-certainties, it is reasonable to think of the location as a probability distributionpeaked at the specified location. If we assume that the location uncertaintiesare Gaussian distributed, we can express an event as a Gaussian probabil-ity density function (PDF) with mean and covariance matrix specified by theevent location and its uncertainty, respectively, i.e.

Pj(x) = k1 j exp[− 1

2(x−x0 j)

T C−1j (x−x0 j)

], (3.1)

28

where Pj(x) is the PDF of event j as a function of the three dimensional spa-tial coordinates x, x0 j is the estimated location, C j is the covariance matrixcontaining information about the spatial uncertainty and k1 j is a normalizationconstant which makes the PDF integrate to unity.

The catalog distribution is then described by the combined probability dis-tribution of all events in the catalog, which is the sum of the probability distri-butions of individual events in the catalog, i.e.

Pc j(x) = k2 j ∑k 6= j

k1k exp[− 1

2(x−x0k)

T C−1k (x−x0k)

], (3.2)

where k2 j represents another normalization constant. This combined probabil-ity distribution contains information about the seismicity in the catalog. Notethat when we calculate the catalog distribution for the jth event, we do not in-clude its PDF. This is because we will use this catalog distribution as a prioriinformation for relocating the jth event.

In the relocation method, the catalog distribution Pc j is used as a priori infor-mation for relocating events in the catalog by Bayesian inversion. Using thecatalog distribution as a priori information is reasonable if one is convincedthat the catalog is complete, i.e. if the existing distribution of events in thecatalog captures all potentially seismogenic regions in the area of the catalog.

3.1.2 The collapsing methodFor the collapsing method, we use the same catalog distribution Pc j as an at-tractor for moving events in an iterative scheme. The attractor pulls the eventsupwards along the local gradient of the catalog distribution. After all events inthe catalog are moved, the catalog distribution is re-calculated and the processis repeated until a termination criterion is met. We choose the termination cri-terion to be the minimum of the misfit between the empirical distribution ofthe normalized displacements of the events and a theoretical chi distributionwith 3 degrees of freedom χ3 (Forbes et al., 2010). Assuming that the locationuncertainties are Gaussian, as we did in the calculation of the catalog distri-bution, random displacement of each event within its Gaussian distribution,normalized by the width of the distribution in the direction of its displacement,follows the χ3 distribution. Therefore, it is justified to continue the iterationto a point where the distribution of normalized displacements approaches a χ3distribution.

Since the collapsing method is not data-driven, i.e. the movements of theevents are guided by the catalog density, but not the data misfit, the collapsedcatalog will have larger data misfits than the original catalog. For a linearized

29

location problem, the sum-of-squares data misfit for a catalog where the nor-malized displacements of the events follow the χ3 distribution is given by

q = q0

(1+

D2

n−m

), (3.3)

where q0 is the misfit in the original catalog, D is the length of the normalizeddisplacements and n and m are the number of observations and the modelparameters, respectively. For a χ3 distribution, D2 = 3 and for the locationproblem, m = 4. Therefore, eq. (3.3) becomes

q = q0

(1+

3n−4

). (3.4)

If we assume, on average, 10 arrival times are used to locate an event, thenthe data misfit will be increased by 50 %, which corresponds to 25 % of thetravel time uncertainty estimates. As mentioned in section 2.4, the locationuncertainties are arguably under-estimated. Therefore, the data misfit is notsacrificed by much by the collapsing.

3.1.3 The May 2008 aftershock sequence in IcelandTo demonstrate our relocation and collapsing methods, we present a real dataexample in Paper I. This example is taken from an aftershock sequence aftera pair of earthquakes in southwest Iceland, which occurred in May 2008. Theaftershock sequence contains more than 18000 events with uncertainties lessthan 3 km in the lateral directions and 6 km in the vertical direction. Theevents are concentrated along three lineations (Figure 3.1).

Figure 3.1. An overview of the 2008 aftershock sequence in southwest Iceland. Theevents (red dots) have estimated location uncertainties of less than 3 and 6 km inthe lateral and the vertical directions, respectively. The black solid lines show thefault lines mapped by Brandsdóttir et al. (2010), Einarsson (2010) and Khodayar andBjörnsson (2014).

30

The north-south striking lineation furthest to the east coincides with the initialearthquake rupture on 29 May 2008. The second north-south lineation slightlyto the west represents the secondary rupture that occurred within seconds ofthe first one (Decriem et al., 2010). The third lineation strikes approximatelyN80E. This fault zone is a part of the complex plate boundary deformation inthe area and appears to have been triggered by the dual event.

In Paper I, we compare our relocation and collapsing methods with the col-lapsing method of Jones and Stewart (1997) and the condensation method ofKamer et al. (2015). The two methods are described in section 2.6. To evalu-ate the effectiveness of different strategies to simplify earthquake catalogs, weuse the entropy to quantify the diffuseness of the earthquake locations, whichis defined as

S =−∫V

P(x) log

[P(x)P0(x)

]dx, (3.5)

where P(x) is the probability density, x is a position vector in the domainV , and log is the natural logarithm. P0(x) is a reference probability density,which is taken to be a uniform probability density such that the entropy of adistribution is zero when it is uniform over its domain. In Paper I, the catalogdistribution, defined as the sum of the probability distributions of individualevents in the catalog, is used as the P(x) in eq. (3.5).

Figure 3.2 shows the results of applying the four strategies to the initial distri-bution of the aftershock sequence. The condensation method has little effecton the event distribution or the event density (Figure 3.2(b)). Our relocationmethod tightens the distribution of events (Figure 3.2(c)). The location uncer-tainties of the events are re-estimated based on the relocated events’ distribu-tion and therefore the uncertainties are reduced. The results of applying theoriginal collapsing method of Jones and Stewart (1997) are shown in Figure3.2(d). The distribution of epicenters is significantly simplified. However, thedistribution looks rather artificial. Initially separated clusters attract each otherand connect. Finally, our collapsing method significantly tightens the distribu-tion of the locations and therefore makes the lineations become more visible(Figure 3.2(e)). It also helps to resolve features in the depth sections of thethree lineations (Figure 3.3).

31

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p.

32

Figure 3.3. The collapsed event locations for the aftershock sequence viewed fromthe south to the north. (a): The original catalog. The upper frame shows the locations(red dots) of the events with error ellipses shown in black. Note that the figure isproduced by projecting all the events onto the x-z plane. The lower frame shows acontour graph of the marginal probability density of the catalog. The dashed linesin both frames indicate the ground level. (b): The same as (a) except showing thecollapsed event locations without error ellipses. In this example, the process stops atthe 4th iteration where the termination criterion is met.

3.2 Summary of Paper IIPaper II focuses on the analysis of the spatial distribution of seismicity in theHengill region, southwest Iceland using the relocation and collapsing methodsintroduced in Paper I and the relocation method developed by Abril and Gud-mundsson (2017) (see section 2.5.1 for a brief description of the method).

The Icelandic Meteorological Office (IMO) has compiled 130,000 events overa 20-year period within the Hengill area. The events in their catalog (SILcatalog) are relocated by application of empirical traveltime tables using anon-linear location strategy (Abril and Gudmundsson, 2017). The relocationsare then redone applying a Bayesian inversion using the catalog event densityas a prior (the relocation method in Paper I). Finally, they are collapsed usingthe same catalog density as an attractor (the collapsing method in Paper I).We show that this way of catalog processing reproduces details of the spatialpattern of seismicity that independently emerges from relative relocations of asmall subset of the catalog events (Bessason et al., 2012). The processed cat-alog is then used to estimate the variable depth extent of the seismogenic zone

33

in the region, associating that with the depth to the brittle-ductile transition(BDT).

3.2.1 Tectonic setting and seismicity of HengillThe Hengill area is a diffuse triple junction where the Western Volcanic Zone(WVZ) and the South Iceland Seismic Zone (SISZ) meet the Reykjanes Vol-canic Zone (RVZ) (Figure 3.4(a)). Three volcanic systems appear in the area(Figure 3.4(b)): 1) The Hengill central volcano and its associated NNE trend-ing fissure swarm, extending from the coast south of Hengill to Lake Þing-vallavatn in the north; 2) Hrómundartindur, to the east of the Hengill sys-tem; 3) Grensdalur, to the south of Hrómundartindur, is extinct and deeplyeroded, but still a source for intense geothermal activity. Several large high-temperature geothermal fields at Nesjavellir, Hellisheiði, Hverahlíð and Ölkel-duháls, are associated with the Hengill and Hrómundartindur volcanoes (Fig-ure 3.4(b)).

Figure 3.4. (a): Map of major tectonic features in Iceland. The orange shaded areasshow the rift zones with the names labeled next to them. The white areas indicatethe glaciers. The red square outlines the study area. The different parts of the plateboundary are labeled with the following abbreviations: RVZ = Reykjanes VolcanicZone, WVZ = Western Volcanic Zone, EVZ = Eastern Volcanic Zone, NVZ = North-ern Volcanic Zone, SISZ = South Iceland Seismic Zone, TFZ = Tjörnes Fracture Zone,RR = Reykjanes Ridge, KR = Kolbeinsey Ridge. (b): An enlarged map of the Hengillarea (red box in (a)) showing the outlines of volcanic centers (dashed) and fissureswarms (solid) using the following abbreviations: He = Hengill, Hr = Hrómundartin-dur, Gr = Grensdalur. The geothermal areas at Nesjavellir (Nv), Hellisheiði (Heh),Hverahlíð (Hvh) and Ölkelduháls (Ökh) are indicated in red.

The Hengill triple junction is an area with high and persistent seismicity. Priorto 1990, about 3000 events were recorded by the permanent Icelandic re-gional seismograph network and some temporary networks deployed in thearea (Foulger, 1988). In August 1994, intense swarm activity started beneath

34

Hrómundartindur volcano, which later spread to the Ölfus area (Figure 3.4(b)).The activity ceased in mid 1996. A second sequence started in 1997 and cul-minated in two magnitude ML = 5.5 earthquakes which occurred in June andNovember 1998. In May 2008, an earthquake doublet with a magnitude of6.3 occurred to the southeast of Hengill. Significant induced seismicity hasoccurred in recent years in association with drilling and fluid injection intothe periphery of the geothermal reservoirs of the Hellisheiði and Nesjavellirpower plants.

3.2.2 Results of relocations and collapsingThree different relocation and collapsing methods are applied sequentially tothe SIL catalog, as described at the beginning of this section.

The catalog after application of each method is shown in Figures 3.5 and3.6. The events in the original catalog are clustered in different areas (Fig-ure 3.5(a)). The biggest cloud in the area between the Hengill and the Hró-mundartindur volcanoes contains more than 50 % of the events in the catalog.Some N-S and ENE-WSW trending lineations are suggested in the cloud. Tothe south of the biggest cloud near to the coast, E-W and N-S trending lin-eations are apparent. The E-W trending lineation seems to be composed ofa number of short N-S trending lineations. To the west of the biggest cloud,there is a cloud of shallow events. This corresponds to the induced seismicityrecorded towards the end of 2011 and into 2012 at Húsmúli.

Figure 3.5. (a): The seismicity of the original SIL catalog. The red dots indicate theevent locations in both map view and depth sections. The depth sections are generatedby projecting all events onto the respective planes. In the map view, the contours ofaltitude are drawn for every 200 m from the sea level. (b): The same as (a), exceptshowing the catalog after relocations with empirical traveltime tables.

35

After relocations and collapsing, the events are clustered and the distributionof the events is tightened (Figures 3.5(b) and 3.6). Lineations become morevisible. They tend to follow two orientations, N-S and ENE-WSW. The depthsections show that the lineations dip nearly vertically.

Figure 3.6. (a): The seismicity of the catalog after relocations using the catalog densityas an a priori constraint. The format of the figure follows that of Figure 3.5. (b): Thesame as (a), except showing the catalog after application of the collapsing method.

The above results are compared to a small subset of the catalog events thatare independently relocated using the method of Slunga et al. (1995). A sum-mary of the relative locations is presented by Bessason et al. (2012). Thecomparison shows that our processed catalog reproduces details of the spatialpattern of seismicity that appears in their relative relocations. Features suchas predominant faulting directions and individual swarm orientations, are wellreproduced. For a figure showing the result of the comparison, the readers arereferred to Figure 5 in Paper II.

3.2.3 Estimating the depth to the brittle-ductile transitionSince the comparison between our processed catalog and the results of rela-tive locations is striking, we argue that our catalog can be used to estimate thedepth to the BDT.

To describe the rheology of the lithosphere, a simple three-layer model withan upper brittle zone and a lower ductile zone is often used. In the brittle zone,the deformation occurs as frictional sliding on fault surfaces and, therefore,they are seismogenic. In the ductile zone, the deformation takes place by bulkplastic flow and are not seismogenic. Between these two zones, there is a BDTwhere deformations are governed by a mix of the two processes. The depth ofthis transition is often interpreted as the base of the seismogenic zone.

36

Figure 3.7. Left frame: (a): The number density of the original SIL catalog. For abetter comparison, the style of the contours is the same as in Figure 3.5. (b): The esti-mated depths to the brittle-ductile transition. Note that estimations using the Gaussianfilter with a standard deviation larger than 2.5 km are masked in this plot. (c): Themap showing the standard deviations of the Gaussian filters used for the estimationsin each cell. Right frame: (d), (e), (f): The same as (a), (b) and (c), except that thedepths to brittle-ductile transition are estimated using the catalog after our relocationand collapsing methods.

In our estimation, the depth to the BDT is defined as the depth above which90 % of the earthquakes are located. Since the events are clustered in space,to avoid bias in the estimation due to limited statistics, an adaptive spatial

37

filter is used, which is a Gaussian distribution around each cell of varyingwidth. Figure 3.7 shows the results of the estimation for the original SILcatalog and the catalog after all relocations and collapsing. In general thedepth to the BDT is shallower in the northern part than the southern part ofthe area. In the southern part, where the seismicity is dominated by tectonicdeformation, the average depth is about 8 km. In the northern part of theregion, where volcanic processes predominate, the average depth is between5.5 and 8 km. The estimates based on the final processed catalog show similarfeatures, except that the depth estimates are shallower.

38

Part II:Tremor locationIn the second part of this thesis, I focus on the topic of tremor location. Thispart is divided into two chapters. In chapter 4, I introduce the basics of thetremor location problem and set the stage for chapter 5, which summariesPaper III and IV.

4. Tremor source location

This chapter presents the background of the tremor location problem. It startswith a description of tremor and a review of existing methods for tremor loca-tion. Then, I focus on the principle of the tremor location by back projection ofcross correlation functions, which is the foundation of the methods describedin Paper III and IV.

4.1 TremorTremor is a continuous record of vibration in the seismograms. Unlike earth-quakes, which manifest themselves as high frequency, short lived signals withdistinct phases in the seismograms, tremor usually has a lower frequency con-tent, much longer duration and a lack of impulsive seismic arrivals. It maycontinue from a few minutes up to several days or weeks. Tremor can be gen-erated by different processes. For example, slow-slip motions at the boundarybetween two tectonic plates may generate tremor that can last for a few weeks(Obara, 2002). Tremor may also be associated with swarms of low-frequencyearthquakes (e.g. Shelly et al., 2007; Frank et al., 2016). In volcanic regions,it is common to observe volcanic tremor related to various processes insidethe volcano (Sgattoni et al., 2017). It is also common to record tremor ingeothermal areas (Gudmundsson and Brandsdottir, 2010), hydrocarbon reser-voirs (Dangel et al., 2003) or localized artificial sources of noise (Latorreet al., 2014).

Based on the origin of the tremor, people usually distinguish between twotypes of tremor, volcanic tremor and nonvolcanic tremor.

4.1.1 Volcanic tremorVolcanic tremor can be defined as the persistent seismic signal (tremor) that isonly observed near a volcanic region. Active volcanoes are sources of a greatvariety of seismic signals. Based on their waveforms, different types of eventsare classified. High-frequency events (or sometimes called A-type events) arevolcanic earthquakes that have waveforms similar to normal shallow tectonicearthquakes (Zobin, 2011). They usually have clear P- and S-phases. Low-frequency events (also called long-period events or B-type events) are earth-quakes that occur at a shallower depth than the high-frequency events (Mi-nakami, 1960). Therefore, surface waves dominate and the S-phases are not

41

clear in the seismograms. They usually occur in swarms and their waveformsand frequency spectra are very similar to those for volcanic tremor. Volcanictremor is a persistent seismic signal in form of an irregular sinusoid, lastingfrom several minutes to several days. It can occur preceding and/or accompa-nying a volcanic eruption.

Volcanic tremor often shows a large temporal variation in amplitude. Theamplitude variation of the tremor in relation to the intensity of the eruptiveactivity is not completely clear. In many cases, an increase of the tremoramplitude can be associated with observations of strong lava fountaining ordome building (e.g. McNutt et al., 1991; Brandsdóttir and Einarsson, 1992;Alparone et al., 2003). However, in other cases, there is no identifiable rela-tionship between the superficial activity and the tremor amplitude. This hasbeen interpreted as a result of variation of magma flow rate at greater depthsin the crust (Gasparini et al., 1992).

Apart from temporal variations, tremor signals also show great variety in theirfrequency contents. For most tremor sources, the energy is concentrated be-tween 0.5 and 7 Hz (Konstantinou and Schlindwein, 2003). The frequencyspectrum usually shows a random distribution of peaks over a large frequencyrange. However, there are also cases where the spectrum only consists of sev-eral distinct and narrow peaks at a fundamental frequency and its harmonics(Schlindwein et al., 1995). This type of tremor is often called harmonic tremor.In some cases, the spectrum can contain only one sharp peak extending over anarrow frequency band (Hurst, 1992). This is called monochromatic tremor.

Polarization analysis has been used to study the wavefield properties of tremorsignals. This type of analysis allows us to identify the types of waves that thesignal is composed of. One standard method is to calculate the covariance ma-trix of signals from a three-component seismograph for a short, sliding timewindow along the time series (Montalbetti and Kanasewich, 1970). Given thetime series from a three-component station, we first bandpass filter the seis-mograms around our frequency of interest. We then calculate the elements ofthe covariance matrix C using

Ci j =1N

N

∑k=1

xi(k) x j(k), (4.1)

where xi(k) is the kth sample of the ith component of the signal, N is the num-ber of samples in one time window, and i, j = Z, N, E represent the vertical,north and east components of the seismograph, respectively.

42

The full covariance matrix for all nine elements is

C =

CZZ CZN CZECNZ CNN CNECEZ CEN CEE

. (4.2)

This matrix can be diagonalized by solving the eigenvalue problem

Cv = λv, (4.3)

where λ and v are the eigenvalues and the eigenvectors of the covariance ma-trix C, respectively. The eigenvectors show the directions of the three principalaxes of the particle motion of the signal, which can be used to infer wave types.Studies have shown that in most cases, volcanic tremor mainly consists of sur-face waves (Ferrazzini et al., 1991; Ereditato and Luongo, 1994; Wegler andSeidl, 1997). However, there are also a few studies showing that there is a mixof surface waves and body waves in the wavefield (Almendros et al., 1997).

4.1.2 Nonvolcanic tremorIn general, nonvolcanic tremor refers to any persistent record of vibration inthe seismogram that is not related to volcanic processes and therefore notfound in a volcanic region. In the stricter context, nonvolcanic tremor canbe defined as tremor that is related to slow-slip, aseismic motion of tectonicplates, mainly found in the subduction zones. In the following discussion, Iwill only concentrate on this type of nonvolcanic tremor.

Nonvolcanic tremor is usually associated with slow-slip events (or slow earth-quakes). Like ordinary earthquakes, slow-slip events are caused by slip onfaults. However, slow earthquakes rupture very slowly, with duration on theorder of days to years. Therefore, they do not generate strong high-frequencyseismic radiation like ordinary earthquakes. For this reason, they have beenreferred to as “silent earthquakes”. Rogers and Dragert (2003) showed thatin northern Cascadia, slow-slip events correlate temporally and spatially withepisodes of nonvolcanic tremor. Each slip event is accompanied by an increaseof tremor activity and this phenomenon repeats regularly with a period of 13- 16 months (Miller et al., 2002). Similar observations were also reported inwestern Shikoku, Japan by Obara et al. (2004), although with a different re-peating cycle due to the difference of conditions on the plate boundary. Thiscoupling of slow-slip events with tremor suggests that the tremor may be trig-gered by the stress perturbation imparted by slow slip (Kao et al., 2006).

Similar to volcanic tremor, nonvolcanic tremor is also highly variable in ampli-tude over time. At many time periods, the tremor can appear to be quite stable,maintaining a nearly constant amplitude over a significant period of time with

43

only little increase or decrease in amplitude. However, at some other times,tremor is rather spasmodic, with many bursts of energy that have significantlyhigher amplitude than the background level. These bursts can last for less thana minute up to tens of minutes.

Nonvolcanic tremor is readily seen in the frequency range of 1 - 10 Hz, withdeficiency of energy at higher frequencies. Although the spectral content ofnonvolcanic tremor can be very similar to that of volcanic tremor, nonvolcanictremor does not appear to be harmonic, a feature that is common for volcanictremor.

The wavefield of nonvolcanic tremor is likely to be dominated by shear wavessince studies have shown that the wavefield propagates at the S-wave veloc-ity (Obara, 2002). This was confirmed by polarization analyses which indi-cate that tremor is mainly composed of shear waves (La Rocca et al., 2005;Miyazawa and Brodsky, 2008; Payero et al., 2008).

4.2 Tremor locationUnlike earthquakes, which often have a clear first arrival pick in the seismo-grams, tremor usually does not have a distinct onset time. Therefore, it isnormally not possible to use picked arrival-time methods, like those describedin chapter 2, to locate tremor sources. Instead, numerous strategies have beendeveloped for this purpose (e.g. Konstantinou and Schlindwein, 2003). Thesemethods usually reproduce more or less the same epicentral locations fortremor, but often have significant differences in their depth estimates. Somemethods suggest that tremor is confined to the interface between two tectonicplates in Japan (Shelly et al., 2006) while other methods indicate that tremoris distributed over a depth range of more than 40 km in Cascadia (Kao et al.,2005). This strong difference in depth requires significantly different physi-cal models to explain tremor generation in these two areas. For this reason,an accurate location of the tremor source in both space and time is a crucialstep to understand the mechanism of tremor generation. Having an accurateand reliable location will allow us to determine whether the differences in thedepth distribution are physical or not and to develop the appropriate model forthe tremor generation.

In the following subsections, I will introduce some of the common methodsfor locating tremor.

44

4.2.1 Amplitude decayAmplitude decay is one of the most common methods for tremor location (e.g.Gottschämmer and Surono, 2000; Patanè et al., 2008). With this method, it isassumed that seismic waves propagate in a homogeneous medium and thereis a simple law that governs the seismic amplitude decay with distance. Thesource location starts with the calculation of the observed amplitudes for allseismic stations. Battaglia and Aki (2003) defined the observed amplitude asthe root-mean-square of the amplitude of a bandpass filtered signal in a timewindow, i.e.

Aobs =

√1N

N

∑i=1

[s(i)]2, (4.4)

where s(i) is the ith sample of the filtered signal in a time window with Nsamples. The predicted amplitude is then computed from an amplitude decaylaw with a general form of

Apre(r, f ) = A0( f )r−αe−β r, (4.5)

where A0 is the reference amplitude of the signal at the source, f is the fre-quency, r is the distance between the source and the station, α is the geomet-rical spreading factor for a specific wave type (body wave or surface wave),and β is a frequency-dependent coefficient that describes the attenuation. Forsurface waves, α = 0.5. The attenuation due to the anelasticity of the mediumis usually described in term of the quality factor Q such that

β =π fQv

, (4.6)

where v is the wave velocity. The source location is then estimated using a gridsearch approach. Since the source amplitude A0 is not known, the search hasto include a search for A0. Therefore, for a two dimensional location problem,the grid will be in three dimensions. For each grid point, Apre is calculated forall stations and the misfit between the predicted and the observed amplitudesis found. The misfit is defined as

E =M

∑i=1

(Ai

pre−Aiobs

)2, (4.7)

where Aipre and Ai

obs are the predicted and the observed amplitudes at the ithstation, respectively, and M is the number of stations. Finally, the best locationis chosen to be the point where the misfit is minimum.

One should note that the amplitude decay law in eq. (4.5) is simplistic. It doesnot allow for focusing/defocusing effects which are expected to be strong at

45

volcanoes because heterogeneity level is expected to be high. It also ignorespossible radiation pattern of the tremor source and possible site amplification,although station corrections are often applied.

Battaglia and Aki (2003) and Di Grazia et al. (2006), for example, appliedthis strategy to locate volcanic tremor at Piton de la Fournaise volcano andMount Etna, respectively.

4.2.2 Semblance methodSemblance is a measure of the coherency in multichannel signals (e.g. Neidelland Taner, 1971) and is given by

S =

N∑j=1

[M∑

i=1fi(τi + j∆t)

]2

MN∑j=1

M∑

i=1fi(τi + j∆t)2

, (4.8)

where τi is the starting time of a time window, ∆t is the sampling interval,fi(τi + j∆t) is the signal recorded at the ith station at time sample j, M is thenumber of stations and N is the number of sample points within the time win-dow. Semblance measures the ratio of the power of the stacked signals thatare aligned for a specific source location to the total power of the data for thewhole array over the time window. If there is a source located at the trial lo-cation, the spatial coherency among the signals radiating from the source willprovide a large value of the semblance. Therefore, by computing the sem-blance for different trial locations, it is possible to map the spatial distributionof the source.

The location is estimated using an approach similar to the grid search methodof Gottschämmer and Surono (2000). First, a reference time ts is fixed. Thisreference time is often chosen to be the first arrival time at a reference stationif the phase is identifiable. For a given trial source location in a two or threedimensional grid, the origin time of that trial source is estimated by

t0 = ts− r/v, (4.9)

where r is the distance between the trial source and the reference station and vis the velocity of the medium. The arrival time at a station i is then calculatedby

τi = t0 + ri/v, (4.10)

where ri is the distance between the trial source and the ith station. This ar-rival time, τi, is used as the starting time of the time window over which the

46

semblance is computed. The spatial distribution of the semblance can then bedepicted where the area with higher values represents the source.

This method was used by Furumoto et al. (1990) and Cannata et al. (2013) tolocate volcanic tremor at Izu-Oshima Volcano in Japan and long-period eventsat Mount Etna in Italy, respectively.

4.2.3 Method of correlation coefficientsAki (1957) developed a method to study the characteristics of a complex wave-field. The method assumes that the wavefield can be described as a stochasticwave which is stationary in space and time. Under this assumption, he formu-lated an equation which expresses the spatial correlation of a stochastic waverecorded at two stations in term of the power spectrum of the wave in the fre-quency domain. This equation provides us a way to identify the predominantwave type and the back azimuth of the incoming waves across an array usingthe spatial correlations. Combining the back azimuths of the waves from dif-ferent arrays allows us to pinpoint a location for the source.

To formulate the strategy, we first define a spatial correlation function betweensignals recorded at two stations as

φ(r,θ) =⟨

u(x,y, t) u(x+ r cosθ ,y+ r sinθ , t)⟩, (4.11)

where 〈...〉 represents temporal averaging, u(x,y, t) is the signal as a functionof time t recorded at location (x, y), r is the distance between two stations andθ is the angle between the x-axis and the line joining the two stations. Takingan azimuthal average, we have

φ(r) =1π

∫π

0φ(r,θ)dθ . (4.12)

Aki (1957) demonstrated that for a single mode stochastic wave with phasevelocity c(ω), where ω is the angular frequency, the azimuthally averagedcorrelation function can be expressed in term of the power spectrum Φ(ω),i.e.

φ(r) =1π

∫∞

0Φ(ω)J0

[ω

c(ω)r]

dω, (4.13)

where J0 is the zeroth order Bessel function of the first kind. Filtering the waveover a narrow band around ω = ω0, the power spectrum is then expressed as

Φ(ω) = Φ(ω0)δ (ω−ω0), (4.14)

47

where Φ(ω0) is the spectral power density at ω0, and δ (ω) is the Dirac deltafunction. Combining eq. (4.13) and (4.14) yields

φ(r,ω0) =1π

Φ(ω0)J0

[ω0

c(ω0)r]. (4.15)

Now, we can define the correlation coefficient as

ρ(r,θ ,ω0) =φ(r,θ ,ω0)

φ(0,θ ,ω0). (4.16)

Taking the azimuthal average of the correlation coefficient, we have

ρ(r,ω0) = J0

[ω0

c(ω0)r]. (4.17)

This equation allows us to determine the phase velocity as a function of fre-quency if we know the correlation coefficients for a fixed r at different fre-quencies. To find the back azimuth of the incoming waves, one can plot acontour map of the correlation coefficients versus azimuth and frequency. As-suming the incoming wave can be described as a plane wave coming from asingle direction θ0, the correlation coefficient becomes

ρ(r,θ ,ω0) = cos[

ω0r cos(θ −θ0)

c(ω0)

]= cos

(ω0∆t

), (4.18)

where ∆t is the differential traveltime between the two stations. The equationshows that if the incoming wave direction is perpendicular to the direction ofthe line joining two stations, the correlation coefficient is a constant. In otherwords, ρ is constant along the wavefront. Therefore, by inspecting the plot ofcorrelation coefficients against azimuth and frequency, one can obtain the backazimuth of the wave. Combining this information from several arrays allowsus to locate the source by triangulation (e.g. Ferrazzini et al., 1991; Métaxianet al., 1997).

4.3 Back projection of cross correlation functionsOver the last decade, the back projection of inter-station cross correlationshas become a common tool for locating tremor (e.g. Haney, 2010; Ballmeret al., 2013). The cross correlation function is a very effective tool to sup-press incoherent random noise in the seismograms and therefore highlightstheir coherency at different time delays. If we assume that the tremor is apoint source which is incoherent in time and the medium between the sourceand all recording seismographs is homogeneous with a constant velocity, then

48

the recorded tremor will consist of a continuous interference pattern betweenthe same source functions at random time intervals, but at a fixed delay forone station compared to any other corresponding to the differential traveltimefrom the source to the two stations. The seismograms of two stations will,therefore, correlate at that time lag, canceling the random time of the sourceimpulses by a relative measurement. This is the principle of using cross cor-relation functions to extract differential traveltimes between station pairs.

Since the cross correlation functions are represented in the time domain, inorder to locate tremor with the cross-correlation method, the simplest way isto back project the cross correlations from the time domain to the spatial do-main, assuming a velocity model. The back projection is usually done in thefollowing manner. First, a geographic grid of potential source locations is de-fined and a velocity model is chosen. For each hypothetical source location,we then calculate the theoretical differential traveltime between two stations.Next, the value corresponding to this time lag is fetched from the cross correla-tion of the two stations. Finally, the maps of back-projected cross correlationsfrom all station pairs are stacked to obtain a map of stacked back-projectedcross correlations which highlights the source location at regions with highervalues.

Figure 4.1. An illustration of the back-projection method for the source location. (a):The back projection of the envelope of the cross correlation between station 1 and2. It is clear that with only one station pair, it is not possible to constrain a sourcelocation. (b): The addition of a second correlation envelope between station 1 and 3gives an intersection, which is the inferred location of the source. (c): The addition ofextra correlation envelopes enhances the intersection relative to the other parts of thehyperbolae. The amplitudes are normalized to the maximum of each frame.

When we back project the cross correlation function of a station pair to thespatial domain, a peak at a given time lag will distribute along a hyperbolicsurface, assuming a uniform velocity in the medium. If we consider two-dimensional space, the peak will distribute along a hyperbola (Figure 4.1(a)).

49

Therefore, with only one station pair, it is not possible to constrain a sourcelocation. Adding an extra station pair will give an extra hyperbola whichusually intersects the existing hyperbola at a point (Figure 4.1(b)). Addingfurther station pairs will enhance that intersection point relative to the otherparts of the hyperbolae (Figure 4.1(c)). Therefore, we need cross correlationsfrom at least two station pairs to locate a source in two dimensions. Simi-larly, we need at least three cross correlations for location in three dimensions.Adding extra station pairs provides redundant information about the sourcelocation and suppresses noise which is inconsistent with a single source solu-tion. However, this kind of suppression is rather weak and is fundamentallylimited by the number of available station pairs. Besides, each back-projectedcross correlation carries a strong signature of the station geometry. When weadd these cross correlations together, we are adding non-equivalent estimatesof the source location.

In chapter 5, I will present a summary of Paper III and IV, which address theseproblems and introduce two new methods for tremor location using higher or-der cross correlations.

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5. Summary of Paper III and IV

In this chapter, I will first give a summary of Paper III, which proposes a newstrategy for tremor location. Then, I will describe the motivation of develop-ing the method proposed in Paper IV and continue to summarize Paper IV.Finally, in the last section, I will discuss a new way of looking into the methodproposed in Paper IV. The results presented in this section are not included inany of the papers.

5.1 Summary of Paper IIITo address the problems of the common cross-correlation method, i.e. weaknoise suppression due to stacking of back-projected correlations and stronggeometrical signature in each back-projected map, we introduce a new strategyfor the back projection, which we refer to as the double-correlation method. Inthe following subsections, I will introduce the method and summarize resultsin the examples.

5.1.1 The double correlation methodAs the name implies, the double-correlation method back projects the correla-tion of two cross correlations from two station pairs for tremor location. Thecross correlation of signals from a pair of stations, a and b, is

Cab( j) =N− j

∑i=1

a(i)b(i+ j), (5.1)

where i and j are the sample indices, a(i) and b(i) are the seismograms atstations a and b, respectively and N is the total number of sample points. Giventhe cross correlation from another station pair c and d,

Ccd( j) =N− j

∑i=1

c(i)d(i+ j), (5.2)

the straightforward way of defining the double correlation is the cross correla-tion of the correlations from these two pairs, i.e.

Cabcd(m) =M

∑j=−M

Cab( j)Ccd( j+m), (5.3)

51

where M = N − 1 is the maximum sample lag in the first cross correlation.Note that the sum in eq. (5.3) will not have much of a contribution fromtime lags beyond a finite limit corresponding to the possible differential trav-eltimes of sources within the network. Therefore, instead of cross correlatingthe whole time series for all possible combinations of station pairs, we imple-ment our double correlation in the following manner.

First, we convert our time series to an analytic signal

f (t) = f (t)+ iH[

f (t)], (5.4)

where f (t) is the original time series, H [...] denotes the Hilbert transform andf (t) is the analytic signal of f (t). Then, we divide the time span of our signalsT into K sub intervals of length ∆T = T/K. Next, we select a station triplet,a, b and c. We choose one of them as a reference, e.g. a. We correlate signalsat stations a and b on one hand and stations a and c on the other. For the kthinterval, the cross correlations at stations a and b on one hand and c and d onthe other, are

kCab( j) =(k+1)∆T

∑i=k∆T+1

a(i) b∗(i+ j) (5.5)

and

kCac( j) =(k+1)∆T

∑i=k∆T+1

a(i) c∗(i+ j), (5.6)

respectively, where ∗ indicates the complex conjugate. Finally, we cross cor-relate the two to obtain the double correlation of the station triplet a, b and c,i.e.

Cabc(m) =K−1

∑k=0

kCab(i) kC∗ac( j+m), (5.7)

where i and j are the expected sample lags calculated from the theoretical dif-ferential traveltimes between stations a and b, and a and c, respectively. Theamplitude of the complex double correlation provides a measure of the doublecorrelation at a given time lag. It will peak at the double differential traveltimebetween two pairs of stations.

The modulus of the double correlation in eq. (5.7) is then back projectedto a geographic grid for source location. Finally, back-projected double corre-lations from all station triplets are stacked.

The use of double correlations addresses our problems in the following two

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ways. First, random noise is better suppressed over the single correlationssince the level of redundancy increases. Second, the back projection of a dou-ble correlation will constrain the location to a point as opposed to a hyperbolafor a single correction. Therefore, we are stacking equivalent estimates of thesource location that do not carry strong signature of the station geometry.

5.1.2 Volcanic tremor at Katla volcano, IcelandIn paper III, the double-correlation method is applied to tremor at Katla vol-cano, Iceland in 2011 for source location (Figure 5.1). 23 hours of signalsfrom 10 stations closest to the caldera are used in the analysis. The signals arecleaned from identifiable earthquakes and filtered between 0.8 and 1.5 Hz (seeSgattoni et al. (2017) for a more detailed description about data preparationand the tremor).

Figure 5.1. A map of Katla volcano. Red inverted triangles show the locations ofseismic stations with their names. The white areas mark glaciers. The green dashedline outlines the caldera.

Figure 5.2 shows a comparison of the 2D back projection of single and doublecorrelations for data with and without one-bit normalization (Li et al., 2017).Compared to the existing single-correlation method, the spurious peaks (peaksthat are consistent with some station pairs, but not all) in the back-projecteddouble correlation are significantly suppressed. The inferred location is muchmore focused compared to that in the single correlation. The inferred locationcoincides with two closely spaced cauldrons that collapsed during the tremorperiod.

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Figure 5.2. (a, b): The back projection of single correlations for tremor at Katla vol-cano, Iceland in July 2011. Two strategies of amplitude normalization are used. Fromleft to right: without any normalization and one-bit normalization of seismograms.(c, d): The same as panels (a) and (b) except showing the back projection of doublecorrelations. The square root of back-projected double correlations is taken to allowa fair comparison to the single correlations. The colors define normalized energy,dark red for maximum energy. Red inverted triangles: seismic stations. Black solidline: glacier. Black dashed line: caldera outline. Black dots: locations of earthquakesoccurred during the tremor period. White open circles: locations of cauldrons.

5.1.3 Synthetic examplesIn the synthetic tests, we try to simulate the real tremor by including vari-ous noise processes and propagation effects in the seismograms. For noiseprocesses we include both incoherent random noise and correlated noise inthe form of the source impulses delayed according to a body-wave velocity.We add signals corresponding to distant plane-wave sources to simulate theeffect of microseisms. For propagation effects we simulate the scattering ef-

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fect by including a number of scatterers inside the caldera. We model theeffect of velocity heterogeneity by delaying the source impulses according toa two dimensional heterogeneous velocity model. The back projection of thesynthetic data shows very similar features to the real data (Figure 5.3). Thespatial distributions of the back-projected energy of both the single and thedouble correlations are well simulated. In the synthetic tests we also test theperformance of the double correlation with different effects added. Our resultsshow that the scattering appears to be the most obscuring effect for recoveringa focused location estimate.

Figure 5.3. (a) and (b): The back projection of single correlations for synthetic exam-ples. Two strategies of amplitude normalization are used. From left to right: withoutany normalization and one-bit normalization of seismograms. (c) and (d): the same as(a) and (b) except showing the back projection of double correlations, again the squareroot is taken for the double correlations. For better comparison the same color scaleis used as the real-data example. The symbols follow Figure 5.2, except that the blackcross denotes the source location.

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In both synthetic and real-data examples, we concentrate our analysis on sur-face waves, i.e. back projecting correlations in 2D with a velocity for surfacewaves. However, this does not require the tremor to be dominated by surfacewaves. The body wave content in the seismograms is considered as correlatednoise, which is well suppressed by the method.

5.2 Higher-order cross correlationsIn Paper III, we demonstrate how the double correlation suppresses randomand correlated noise and therefore provides a more robust estimate of thetremor location. Since the double correlation provides better noise suppres-sion than the single correlation, one could then ask if it is possible to furthersuppress noise by going to higher-order cross correlations. A simple answerto that is yes. However, the computation of higher-order correlations becomesmore tedious when the order increases. In Paper IV, we propose a strategywhich resembles higher-order correlation. But, it is much less computation-ally expensive than the computation of the full cross correlation. We note thatfor a specific time lag that corresponds to the double differential traveltimebetween two station pairs, the product of the two cross correlations at thattime lag is equivalent to their cross correlation at the same lag. Therefore, it ispossible to use the product of cross correlations to construct the higher-ordercorrelation for a specific time lag.

In an area where the velocity structure is complex and heterogeneous, if wehave imperfect information about this complex velocity structure, the pre-dicted time lags will be imprecise. In addition, signals at different stationsmay be significantly different due to, e.g., propagation complexity. Using theenvelopes of cross correlations may help to stabilize the estimate, e.g., withrespect to the uncertainty of the traveltime prediction in a 2D or 3D veloc-ity field since the envelopes are smoother than the correlations and they areless sensitive to lack of precision in the predicted traveltime and waveformvariation. Therefore, in Paper IV, we use the envelopes of cross correlations,instead of the actual correlations, to compute the products.

5.3 Summary of Paper IVIn Paper IV, we introduce a tremor location method which resembles higher-order cross correlation, but is less computationally expensive than calculatingthe full cross correlations. This method is composed of a spectrum of meth-ods which raises from the stack of correlation envelopes to their highest-orderproducts. We refer to them by the order of the involved products. By secondorder, we mean the product of two correlation envelopes. By third order, we

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mean the triple product of correlations from three station pairs. The followingsubsections summarize the method and results in the examples.

5.3.1 Tremor location using products of correlation envelopesThe location process is achieved by back projecting the higher-order products.Therefore, we first have to define a way to compute the higher-order products.

To compute a second-order product of correlation envelopes, we first selecta triplet of stations, a, b and c. We choose one of them to be a referencestation, e.g., station a. We then compute the correlation envelopes for bothstations a and b and stations a and c. For a hypothetical source location x inthe geographic grid, the differential traveltimes between stations a and b and aand c are calculated assuming a velocity model. We fetch the values that corre-spond to these differential traveltimes from the correlation envelopes. Finally,we multiply these two values together to obtain the second-order product forthat hypothetical location x, i.e.

C2abc(x) =Cab(tx

ab) Cac(txac), (5.8)

where C2abc(x) is the second order product at location x, Cab(tx

ab) and Cac(txac)

are the correlation envelopes between station pairs a and b, and a and c at ex-pected time lags tx

ab, and txac, respectively. The above procedure is repeated for

all the grid locations to obtain a map of a back-projected second-order productof correlation envelopes. Finally, the maps for different triplets of stations arestacked.

Following a similar procedure as the calculation of the second-order prod-ucts, one can compute the higher-order products by including more stations inthe group. For example, when we calculate the fifth-order product, we selectsix stations from the network and choose one of them as the reference station.This way of combining stations only allows us to calculate products up to themaximum (n− 1)th order, assuming that there are n stations in the network.The number of combinations for a group of m stations is given by

N = m(

nm

). (5.9)

The major advantage of using the product of correlation envelopes for tremorlocation is that the correlated noise that remains after the first cross correlationcan be effectively suppressed. This is explained in detail in Paper IV. In thefollowing paragraphs, I will highlight the main points.

When discussing noise suppression, it is useful to distinguish between ran-dom and correlated noise. In the first step of our method, we cross correlate

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seismograms. In the examples presented in Paper IV, the time series are 9hours long. This corresponds to about 20000 degrees of freedom in each timeseries. Therefore, random noise is likely to be strongly suppressed after thefirst correlations. The noise that remains in the data is then dominated by cor-related noise.

Many effects can contribute to the correlated noise. In general, any peaks inthe correlations that do not correspond to the differential traveltimes betweenthe primary source and the stations can be regarded as correlated noise. In ourmethod this noise is suppressed by the products and the sums, not only by thesums.

We can understand how the products and the sums suppress correlated noisebetter than the sums alone with the following conceptual example. Considera case where we have a single source and only three cross correlations. Asdiscussed in the previous section, a peak in the cross correlation outlines a hy-perbola in the spatial domain, assuming uniform velocity in two dimensions.The three hyperbolae will intersect at the true source location, and will, there-fore, give a large value at this location, either by summation or multiplication.Any two hyperbolae will in general also intersect at a trial location and eithersummation or multiplication will produce a large value. However, it is un-likely that the third hyperbola will pass through that point. Therefore, it willgive a small value at this location. If this small value is added to the sum ofthe other two crossing hyperbolae, it will barely affect the result. But, if wemultiply the values from all the three correlation at this location, the result willbe small. In other words, a product is better at suppressing the spurious peaksthan summation.

5.3.2 Synthetic and real-data examplesIn Paper IV, we apply the proposed method to the same tremor (tremor at Katlavolcano in 2011) as was used by the double-correlation method. The results ofthe 2D back projection show that the primary peak in the back-projected mapsbecomes more focused as the order of the product increases (Figure 5.4). Spu-rious energy (hyperbolae corresponding to constant time shifts for the threelinearly aligned stations near the northeast rim of the caldera) is graduallysuppressed as the order increases. In the highest sixth-order products, the pri-mary peak coincides with locations of ice cauldrons. Several secondary peaksare interpreted to be effects of strong scattering in the area.

Apart from the tremor at Katla volcano, the method is also applied to syntheticdata. The effects included in the synthetic data are similar to those added tothe data for the synthetic tests of the double-correlation method. We generate

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the data based on a heterogeneous velocity model. We simulate the presenceof body waves and the effects due to scattering. We include random noise inthe seismograms.

Figure 5.4. (a-f): The back projection of stacked products of correlation envelopes fortremor at Katla volcano, Iceland, on 8 July 2011. The order of the product is indicatedat the top left corner of each frame. To allow a fair comparison between differentorders, the nth root is taken for each back-projected map, where n is the order of theproduct. The colors define the normalized energy. Red inverted triangles show thelocations of the seismic stations. The solid and dashed lines outline the glacier and thecaldera, respectively. In (a) red open circles indicate the positions of cauldrons andblack dots show the seismicity during the tremor period.

In the synthetic tests, two sources, with equal amplitude, are used. One ofthem is placed outside the caldera in order to demonstrate the method’s abilityto resolve sources outside the scattering region (the area within the caldera).For the back projection, we use a slightly higher velocity than the mean ve-locity of the true velocity model in order to study the effect of using a wrongvelocity.

The tests are divided into two parts. To demonstrate the effect of noise sup-pression using higher-order products, we first exclude the scattering effects.The results show how the spurious peaks are suppressed gradually as the orderincreases (Figure 5.5(a)). The energy peaks in the sixth-order product coin-cide with the true locations of sources. Despite the use of a slightly incorrectvelocity, we are still able to recover sources with a resolution of ∼ 1 km. To

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simulate the features in the real data, we then include the scattering effects.The results are similar to those in the real-data example where primary peakscoincide with the true source locations and secondary peaks exist due to thescattering (Figure 5.5(b)). This implies that significant scattering is present atKatla.

Figure 5.5. Back projection results of stacked products of correlation envelopes forsynthetic examples (a) without scattering and (b) with scattering. The figure format isthe same as Figure 5.4. Black crosses denote synthetic source locations.

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5.4 Tremor location using a probabilistic approachAs introduced in section 4.3, when we back project the envelope of the crosscorrelation of signals from a station pair, a peak at a given time lag yields a hy-perbola in two dimensions, assuming a uniform velocity. This back-projectedspatial distribution of an individual cross-correlation envelope can be relatedto a probability distribution of the source location, or a likelihood functionin a Bayesian context, i.e. the probability density of data given a specificmodel location. This is because wherever the amplitude of the back-projectedcorrelation envelope is high, a source location is likely and vice versa. It is,however, not the exact probability distribution in a strict sense because it isunclear how the location uncertainties are related to the width of the back-projected correlation envelope. However, it is reasonable to expect that theprobability distribution or the likelihood function of the source location is amonotonically increasing function of the back-projected correlation-envelopedistribution. The individual back-projected correlation envelopes are indepen-dent because they are independently derived from the data. Therefore, theindividual likelihood functions are independent and the joint likelihood of allcorrelation data is simply the product of the individual likelihood functions.The joint likelihood will be approximated by the product of back-projectedcorrelation envelopes because of their monotonic relationship with the indi-vidual likelihood.

Now, it becomes clear that the product-of-envelope approach described in sec-tion 5.3.1 can be viewed from a different perspective, a probabilistic perspec-tive where individual source locations are estimated from the joint probabilitydistribution of data given a specific model (the velocity). Drew et al. (2013)chose a similar approach when locating micro earthquakes based on back pro-jecting detection attributes such as a short-term-average to long-term-average(STA/LTA) amplitude distribution in time.

To implement this probabilistic approach, we follow a similar procedure asdescribed in section 5.3.1. However, instead of multiplying the back-projectedcorrelation envelopes sharing a common reference station, we propose multi-plying all the available back-projected correlation envelopes together. This isconsistent with the argument that each back-projected correlation envelope isa likelihood function derived from independent data and the joint probabilitydistribution is the product of all likelihood functions.

Assume that we have a network of n seismographs, the number of availablecorrelation envelopes is then

(n2

). For each hypothetical source location x in a

geographic grid, we calculate the theoretical traveltime differences for all(n

2

)pairs of stations. We then fetch the values corresponding to the differentialtraveltimes from the correlation envelopes. Finally, we multiply all the values

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together to obtain the product of all correlation envelopes for that hypotheticallocation x, i.e.

Cprob(x) =n−1

∏i=1

n

∏j=i+1

Ci j(txi j), (5.10)

where Cprob(x) is the product of correlation envelopes defined under the prob-abilistic approach, Ci j(tx

i j) is the correlation envelope between stations i and jat expected time lag tx

i j.

We also apply this probabilistic approach to tremor at Katla volcano, Iceland.The results of the 2D back projection show that the estimated location is highlyfocused on a small area to the southeast of the caldera (Figure 5.6(b)). Anenlarged map of this area (Figure 5.6(c)) suggests that the back-projected en-ergy is concentrated on two separated, but closely located peaks. These peaksare located close to two cauldrons (white open circles in Figure 5.6(c)) thatwere deepened during the tremor period. This lends credibility to the locationmethod. Figure 5.6(d) shows a logarithmic plot of the back-projected map inFigure 5.6(b). The color scale in the plot is normalized to span over five ordersof magnitude from the maximum energy, i.e. the areas with value 0 are fiveorders of magnitude smaller than those with value 1.

Since the probabilistic approach gives a strong suppression on correlated noise,one may attempt to conclude that the probability approach should be preferredfor tremor location. However, one should note that this strong suppression ofcorrelated noise implies that only the primary source will be located. If thereexists a secondary weaker source, it will not be detected with this approach.

Although we build the probabilistic approach on the argument that the prob-ability distribution or the likelihood function of the source location is likelyto be a monotonically increasing function of the back-projected correlation-envelope distribution, it is unclear what exactly the mapping is. Besides,one should note that the back-projected maps obtained from the product-of-envelope approach should not be directly compared to those obtained from theprobabilistic approach since the maps obtained from the product-of-envelopeapproach have a dimension of energy (nth root of the product) while the mapsfrom the probabilistic approach have a dimension of energy to the nth power,where n is the number of correlations being multiplied.

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Figure 5.6. (a): The back projection of stacked correlation envelopes for tremor atKatla volcano, Iceland, on 8 July 2011. (b): The back projection of the product of allavailable correlation envelopes. (c): An enlarged map of the area that corresponds tothe peaks in (b). White open circles mark the locations of cauldrons. (d): A logarith-mic plot of (b). The color scale in this frame is normalized to span over five ordersof magnitude from the maximum energy. For other frames, the colors define the nor-malized energy. Red inverted triangles show the locations of the seismic stations. Thesolid and dashed lines outline the glacier and the caldera, respectively.

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6. Sammanfattning på svenska (Summary inSwedish)

Jordbävningar kan vara destruktiva och skrämmande. Därför är det en viktiguppgift för seismologer att studera dem, beskriva riskerna som de innebär ochförsöka förutsäga dem. Men, jordskalv är mycket mer än bara det skrämman-de fenomen som allmänheten hör och läser om i media. Faktiskt är det så attbara 10 av varje miljon skalv som seismologer registrerar med sina instrumentorsakar skador. De absolut flesta är små och uppfattas inte av någon männi-ska. Alla dessa små, ofarliga skalv skapar olika typer av vågor, bland demtryckvågor av samma typ som ljud. Det är dessa som registreras av seismo-logernas seismografer, det vill säga jordskalvens röster. Deras röster berättarför seismologerna var och hur de blev till och vad som har hänt dem på vägenmot seismografen. Det finns därför mycket mer för seismologer att göra änatt studera samhällsrisker som orsakas av seismiska fenomen. En annan hu-vuduppgift är att tyda jordskalvens berättelser om processer och varierandestruktur inuti jorden.

Seismologernas instrument registrerar fler fenomen än jordskalv. Kontinuer-ligt skakningsbrus kan uppstå på grund av körande bilar, gående människor,vind och havsvågor o.s.v. Processer inuti jorden kan också orsaka kontinuer-ligt skakningsbrus som brukar kallas för tremor. Till exempel brukar vulkanerskaka jämt under sina utbrott.

Det är inte alltid lätt för seismologerna att tyda det som jordens röster försökerberätta för dem, eller så går det inte att översätta hennes språk tillräckligt no-ga för att få tag i tillfredsställande uppgifter. Därför jobbar seismologer helatiden med metodutveckling för att kunna undersöka fler detaljer om jordensinre processer och struktur.

Denna avhandling fokuserar på metodutveckling i två olika sammanhang sombåda förknippas med lokalisering av naturliga seismiska källor. Den introdu-cerar nya strategier för lokaliseringen och ger exempel på applicering av dessastrategier. Lokalisering av seismiska källor är ett viktigt ämne inom seismolo-gin. Oftast är det den första uppgiften när elastiska vågor från naturliga käl-lor analyseras. En precis lokalisering är en förutsättning för vidare analys avvågorna. Två klasser av seismiska källor kan definieras utifrån de registrera-de vågorna. Den första, källor med en tydlig början, t.ex. jordbävningar och

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explosioner. Den andra, källor utan identifierbar början, som tremor och jord-skalv med gradvis början. Lokaliseringen av dessa två typer av källor kräverolika strategier.

Avhandlingen består av två delar. I den första delen diskuteras lokalisering-en och omlokaliseringen av källor med tydlig början, d.v.s. jordskalv. Två nyastrategier för omlokalisering av skalv registrerade i en katalog föreslås. Denförsta strategin använder samlade uppgifter om de registrerade skalven somen “a priori” restriktion av en Bayesisk inversion. De samlade uppgifterna ut-trycks i termer av den kombinerade sannolikhetsfördelningen av alla skalv ikatalogen. I den andra strategin används samma fördelning som en magnetsom drar till sig skalvens lokaliseringar genom en upprepad process. Dessatvå strategier appliceras till efterskalven efter ett par jordbävningar i sydvästraIsland under 2008 och en katalog från SIL nätverket på Island från 1993 till2012. De omlokaliserade skalven i den senare katalogen används för att nogaestimera hur djupt jordskalv förekommer i Hengill området i sydvästra Islandoch hur detta djup varierar. Vi har lyckats demonstrera att efterskalven i syd-västra Island under 2008 fördelas mycket snävt i området runt huvudskalvensförkastningar och har därmed kunnat noga definiera förkastningarnas geome-tri. Vi har påvisat att jordskalvens djup i Hengill området varierar en del. I densydliga delen av området, som främst påverkas av tektoniska processer, nårskalven ett djup på 8 km medan i den nordliga delen, som är vulkanisk, når debara till ett djup på 5-6 km. Detta är bevis för att jordskorpan i det vulkaniskaområdet är väsentligt varmare på djupet, där vi inte kan komma åt den medborrhål, eftersom mycket varma bergarter inte kan knäckas utan istället är fly-tande.

I avhandlingens andra del föreslås två metoder för lokalisering av källor förtremor. Dessa metoder använder en specifik hastighetsmodell för att projekte-ra korskorrelationer av data insamlat på två olika mätstationer från tid till rum.Dock, istället för att projektera korrelationer används dubbelkorrelationer i denförsta metoden. Dessa definieras som korskorrelationer av korrelationerna fråntvå par stationer som har en gemensam referensstation. I den andra metodenanvänds istället produkten av korskorrelationernas amplituder från en gruppstationer med en gemensam referensstation. Dessa två metoder appliceras påvulkanisk tremor från den Isländska vulkanen Katla under 2011. Resultatenvisar att både slumpvis och korrelerat brus dämpas mer med dessa metoderjämfört med vanliga, enkla korrelationsmetoder. Påföljande förbättring av lo-kaliseringen av tremorn från Katla har lett till nya insikter om dess ursprung.Tremorns primära källa var stabilt lokaliserat över tid på ett ställe som förknip-pas med hastig smältning av överliggande glaciär (där djupa grytor förekom iglaciärens yta) och måste därför vara vulkanisk eller geotermisk. Mindre delarav tremorn kan förknippas med smältvattnets färd under glaciären mot mindrehöjd som till slut ledde till översvämningar på låglandet sydost om vulkanen.

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7. Acknowledgments

Over the past few years, I was always imagining how I would feel when Ireached the point that I started writing this acknowledgment. Even now, I stillremember the time when I saw an email from Ari, inviting me to his officeand discussing the opportunity to apply for this PhD position. I have to admitthat I did not think too much at that time about where to pursue a PhD. So,I decided to apply for the position without much hesitation. Now, five yearshave passed, all I can say is that I did make a correct choice. I am so thankfulthat over these years, I could work and discuss problems with groups of intel-ligent people around me and learn so many things from them. Working withthem allowed me not only to finish my thesis, but also to grow and to developto become an independent researcher. I would like to take this moment andexpress my gratitude to all of them.

My deepest gratitude goes first and foremost to my supervisors, Ólafur Gud-mundsson, Ari Tryggvason and Roland Roberts. Oli, you are a very kind andpatient supervisor. Every time when I came to your office with a bunch ofproblems, you were willing to invest your time in patiently answering all ofmy questions. Your broad knowledge in different aspects of geophysics broad-ened my horizons and your clear mind in explaining concepts was a great rolemodel for me. I am so grateful to have you as my supervisor. Ari, thank youfor your encouragement to me to pursue this PhD and also your guidance andsupport during the earlier part of my studies. Roland, thank you for your criti-cal comments on my research and your innovative ideas about locating sourcesusing high-order correlations. Your comments helped me to develop criticalthinking skills on scientific issues. Your ideas inspired me and gave me thechance to step into the field of volcano seismology.

I am also greatly indebted to Hamzeh Sadeghisorkhani. I am glad that wediscussed, explored and learned so many things together during these years.Thank you for sharing your interest on different scientific topics with me, es-pecially your project on ambient noise. I will never forget the time we spentin the office, talking about politics, religions and Iranian culture.

I would like to extend my sincere gratitude to Giulia Sgattoni. With yourfully processed Katla data, I was lucky that I did not have to deal with thosemessy data at all. It would not have been possible but for the contributionsof Claudia Abril. Thank you for relocating the SIL catalog using your new

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strategy so that I had greatly improved data to start with when I applied myrelocation method on the SIL catalog. I also owe my sincere gratitude to PeterHedin for being my mentor and also providing me with lots of teaching mate-rials for the Global Geophysics course. I was glad that I did not have to startfrom scratch for preparing assignments and lecture notes.

I would like to express my sincere gratitude to my colleagues in the seis-mology group, Angeliki Adamaki, Samar Amini, Karin Berglund, DarinaBuhcheva, Zeinab Jeddi and Frederic Wagner. It is my pleasure to meet allof you. I will never forget the time we spent together during conferences, theexciting field trip in Iceland and the little surprise on my 25th birthday.

Many thanks go to the administration staff in our department, particularly Fa-tima Ryttare-Okorie, who helped me a lot with the tedious paper works at thestart of my PhD studies. She has always been working behind us in order togive us the best support.

I am most indebted to all of my co-authors for the constructive comments.Ólafur Gudmundsson, Ari Tryggvason, Reynir Bödvarsson, Bryndís Brands-dóttir, Claudia Abril, Giulia Sgattoni, Hamzeh Sadeghisorkhani and RolandRoberts, without the contributions of all of your comments, it would not bepossible for me to have great improvement in the manuscripts.

I am deeply grateful for the help of Ólafur Gudmundsson and Gudlaug Olafs-dottir, who contributed a lot in the Swedish summary of this thesis.

High tribute shall be paid to my teachers, Bjarne Almqvist, Christoph Hi-eronymus, Christopher Juhlin, Björn Lund, Alireza Malehmir, Laust Pedersenand Peter Schmidt for the knowledge that I gained from their lectures.

Special thanks go to all the people who made my life in Sweden very de-lightful, Amy Au, Joyce Chan, Ne Xun Chan, Anders Cheung, Cecilia Che-ung, Ricky Cheung, Jonathan Forman, Jonas Holm, Hemanth Kumar, MalinLindqvist, Johnny Ragnarsson, Constance Tang, Sally Usher, Jessie Wong,Qian Yu and Wenjun Zhao.

Finally, I would like to take this opportunity to express my deepest gratitudeto my beloved family, especially to my parents, who have always been therefor me. Your never-ending love and care is my greatest power to overcomeevery challenge in these years. In addition, I would like to offer my heartfeltgratitude to my beloved wife, Kwan Yin Lau. Thank you for your unlimitedand selfless supports in every moment of my PhD. Words cannot describe howconsiderate you have been and how much you have sacrificed in order to fullysupport me to pursue my dream in these years.

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References

Abril, C., and Ó. Gudmundsson (2017), Empirical travel time tables for the SILnetwork in Iceland and consequent relocations, paper in preparation.

Aki, K. (1957), Space and time spectra of stationary stochastic waves, with specialreference to microtremors, Bulletin of the Earthquake Research Institute,University of Tokyo, 35(3), 415–456.

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